A particle is moving along the $x-$axis with its coordinate with the time '$t$' given be $\mathrm{x}(\mathrm{t})=10+8 \mathrm{t}-3 \mathrm{t}^{2} .$ Another particle is moving the $y-$axis with its coordinate as a function of time given by $\mathrm{y}(\mathrm{t})=5-8 \mathrm{t}^{3} .$ At $\mathrm{t}=1\; \mathrm{s},$ the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{\mathrm{v}} .$ Then $\mathrm{v}$ (in $\mathrm{m} / \mathrm{s})$ is

A particle is moving eastwards with a speed of $6 \,m / s$. After $6 \,s$, the particle is found to be moving with same speed in a direction $60^{\circ}$ north of east. The magnitude of average acceleration in this interval of time is ....... $m / s ^2$

Two projectiles are thrown simultaneously in the same plane from the same point. If their velocities are $v_1$ and $v_2$ at angles $\theta _1$ and $\theta_2$ respectively from the horizontal, then answer the following question

The trajectory of particle $1$ with respect to particle $2$ will be

and direction of the vectors $\hat{ i }+\hat{ j }$, and $\hat{ i }-\hat{ j }$ ? What are the components of a vector $A =2 \hat{ i }+3 \hat{ j }$ along the directions of $\hat{ i }+\hat{ j }$ and $\hat{ i }-\hat{ j } ?$

The figure shows a velocity-time graph of a particle moving along a straight line  The correct displacement-time graph of the particle is shown as